00:01
The first term of a geometric sequence with a non -zero common ratio is one.
00:05
The first term of an arithmetic sequence is zero.
00:08
The two sequences have the same second term and the fourth term of the geometric sequence equals the tenth term of the arithmetic sequence.
00:14
Find the possible values for the common ratio of the geometric sequence and the corresponding values for the common difference in the arithmetic sequence.
00:23
So, we're going to set up our, set up a table for our arithmetic sequence.
00:29
So, we'll let this be the column of the nth term and we'll let this be g of n.
00:37
We'll set that up as a function, a geometric sequence with respect to n.
00:41
So, we know the first term, when n equals one, that first term is one, given to be one.
00:49
The second term would be found by multiplying that previous term by the common ratio.
00:56
So, it would be one times r.
00:58
The third term would be found by multiplying that one r term by r, which would give us r squared.
01:05
And then the fourth term would be r squared times r, which would be one r cubed.
01:10
And we could continue that pattern on.
01:12
And then, if we get a function that defines that, that would be the nth term.
01:21
We would have one times n.
01:24
And we can see the exponent's always one less than the nth term.
01:28
So, that would be one r.
01:35
And then the exponent is always one less than the nth term.
01:39
So, that would be n minus one.
01:41
So, it gets a formula for the nth term of the geometric sequence with the common ratio of r.
01:47
So, next, let's do the arithmetic sequence.
01:49
N, and we'll let this be a as a function of n for the arithmetic sequence.
01:56
We know that its first term is zero.
01:58
So, the first term is zero.
02:00
And we also know that the two sequences have the same second term.
02:07
So, the second term, it was one r for the geometric sequence.
02:14
It has to be the same here.
02:16
That would be also one r in the arithmetic sequence.
02:20
So, that shows our common difference of this.
02:23
The difference, the common difference would be the difference between each value.
02:27
From zero to one r, that common difference is just r...